Place chicken breasts between plastic wrap and firmly pound with mallet until approx 1/4" thick. Cover and chill while making pesto.
In blender or food processor, combine basil, cheese, oil and garlic and whirl until it becomes a thick paste.
Spoon equal portions of pesto onto each chicken breast. Roll up each breast to enclose pesto and secure with toothpick.
In 12-14" fry pan, melt butter in oil over medium-high heat. Dip each chicken roll in flour, shake off excess, and add to pan. Cook, turning as needed, until golden brown and meat is opaque in center, 6 to 8 minutes.
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